Abstract
We introduce a refinement of the GPY sieve method for studying prime $k$tuples and small gaps between primes. This refinement avoids previous limitations of the method and allows us to show that for each $k$, the prime $k$tuples conjecture holds for a positive proportion of admissible $k$tuples. In particular, $\liminf_{n}(p_{n+m}p_n)<\infty$ for every integer $m$. We also show that $\liminf(p_{n+1}p_n)\le 600$ and, if we assume the ElliottHalberstam conjecture, that $\liminf_n(p_{n+1}p_n)\le 12$ and $\liminf_n (p_{n+2}p_n)\le 600$.

[ElliottHalberstam] P. D. T. A. Elliott and H. Halberstam, "A conjecture in prime number theory," in Symposia Mathematica, Vol. IV, London: Academic Press, 1970, pp. 5972.
@incollection {ElliottHalberstam, MRKEY = {0276195},
AUTHOR = {Elliott, P. D. T. A. and Halberstam, H.},
TITLE = {A conjecture in prime number theory},
BOOKTITLE = {Symposia {M}athematica, {V}ol. {IV}},
VENUE={{INDAM},
{R}ome, 1968/69},
PAGES = {5972},
PUBLISHER = {Academic Press},
ADDRESS = {London},
YEAR = {1970},
MRCLASS = {10.64},
MRNUMBER = {0276195},
MRREVIEWER = {K. Thanigasalam},
zblnumber = {0238.10030},
} 
[FriedlanderGranville] J. Friedlander and A. Granville, "Limitations to the equidistribution of primes. I," Ann. of Math., vol. 129, iss. 2, pp. 363382, 1989.
@article {FriedlanderGranville, MRKEY = {0986796},
AUTHOR = {Friedlander, John and Granville, Andrew},
TITLE = {Limitations to the equidistribution of primes. {I}},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {129},
YEAR = {1989},
NUMBER = {2},
PAGES = {363382},
ISSN = {0003486X},
CODEN = {ANMAAH},
MRCLASS = {11N05},
MRNUMBER = {0986796},
MRREVIEWER = {S. W. Graham},
DOI = {10.2307/1971450},
zblnumber = {0671.10041},
} 
[GGPY] D. A. Goldston, S. W. Graham, J. Pintz, and C. Y. Yildirim, "Small gaps between products of two primes," Proc. Lond. Math. Soc., vol. 98, iss. 3, pp. 741774, 2009.
@article {GGPY, MRKEY = {2500871},
AUTHOR = {Goldston, D. A. and Graham, S. W. and Pintz, J. and Yıldırım, C. Y.},
TITLE = {Small gaps between products of two primes},
JOURNAL = {Proc. Lond. Math. Soc.},
FJOURNAL = {Proceedings of the London Mathematical Society. Third Series},
VOLUME = {98},
YEAR = {2009},
NUMBER = {3},
PAGES = {741774},
ISSN = {00246115},
MRCLASS = {11N25 (11N36)},
MRNUMBER = {2500871},
MRREVIEWER = {Frank Henry Thorne},
DOI = {10.1112/plms/pdn046},
zblnumber = {1213.11171},
} 
[GPY:ManyPrimes] D. A. Goldston, J. Pintz, and C. Y. Yildirim, "Primes in tuples. III. On the difference $p_{n+\nu}p_n$," Funct. Approx. Comment. Math., vol. 35, pp. 7989, 2006.
@article {GPY:ManyPrimes, MRKEY = {2271608},
AUTHOR = {Goldston, Daniel A. and Pintz, J{á}nos and Yıldırım, Cem Yal{ç}ım},
TITLE = {Primes in tuples. {III}. {O}n the difference {$p\sb {n+\nu}p\sb n$}},
JOURNAL = {Funct. Approx. Comment. Math.},
FJOURNAL = {Uniwersytet im. Adama Mickiewicza w Poznaniu. Wydzia\l Matematyki i Informatyki. Functiones et Approximatio Commentarii Mathematici},
VOLUME = {35},
YEAR = {2006},
PAGES = {7989},
ISSN = {02086573},
MRCLASS = {11N05 (11N13)},
MRNUMBER = {2271608},
MRREVIEWER = {S. W. Graham},
DOI = {10.7169/facm/1229442618},
zblnumber = {1196.11123},
} 
[GPY] D. A. Goldston, J. Pintz, and C. Y. Yildirim, "Primes in tuples. I," Ann. of Math., vol. 170, iss. 2, pp. 819862, 2009.
@article {GPY, MRKEY = {2552109},
AUTHOR = {Goldston, Daniel A. and Pintz, J{á}nos and Yıldırım, Cem Y.},
TITLE = {Primes in tuples. {I}},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {170},
YEAR = {2009},
NUMBER = {2},
PAGES = {819862},
ISSN = {0003486X},
CODEN = {ANMAAH},
MRCLASS = {11N05 (11N36)},
MRNUMBER = {2552109},
MRREVIEWER = {S. W. Graham},
DOI = {10.4007/annals.2009.170.819},
zblnumber = {1207.11096},
} 
[GoldstonYildirim] D. A. Goldston and C. Y. Yildirim, "Higher correlations of divisor sums related to primes. III. Small gaps between primes," Proc. Lond. Math. Soc., vol. 95, iss. 3, pp. 653686, 2007.
@article {GoldstonYildirim, MRKEY = {2368279},
AUTHOR = {Goldston, D. A. and Yıldırım, C. Y.},
TITLE = {Higher correlations of divisor sums related to primes. {III}. {S}mall gaps between primes},
JOURNAL = {Proc. Lond. Math. Soc.},
FJOURNAL = {Proceedings of the London Mathematical Society. Third Series},
VOLUME = {95},
YEAR = {2007},
NUMBER = {3},
PAGES = {653686},
ISSN = {00246115},
MRCLASS = {11N05 (11N37)},
MRNUMBER = {2368279},
MRREVIEWER = {D. R. HeathBrown},
DOI = {10.1112/plms/pdm021},
zblnumber = {1118.11040},
} 
[Polymath] D. H. J. Polymath, New equidistribution estimates of Zhang type, and bounded gaps between primes.
@misc{Polymath,
author={Polymath, D. H. J.},
TITLE={New equidistribution estimates of {Z}hang type, and bounded gaps between primes},
NOTE={preprint},
ARXIV = {1402.0811},
} 
[Selberg:CollectedWorks] A. Selberg, Collected Papers. Vol. II, New York: SpringerVerlag, 1991.
@book {Selberg:CollectedWorks, MRKEY = {1295844},
AUTHOR = {Selberg, Atle},
TITLE = {Collected Papers. {V}ol. {II}},
NOTE = {with a foreword by K. Chandrasekharan},
PUBLISHER = {SpringerVerlag},
YEAR = {1991},
PAGES = {viii+253},
ISBN = {3540506268},
MRCLASS = {01A75},
MRNUMBER = {1295844},
MRREVIEWER = {Matti Jutila},
ADDRESS = {New York},
zblnumber = {0729.11001},
} 
[Zhang] Y. Zhang, "Bounded gaps between primes," Ann. of Math., vol. 179, iss. 3, pp. 11211174, 2014.
@article {Zhang, MRKEY = {3171761},
AUTHOR = {Zhang, Yitang},
TITLE = {Bounded gaps between primes},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {179},
YEAR = {2014},
NUMBER = {3},
PAGES = {11211174},
ISSN = {0003486X},
MRCLASS = {11N05 (11N36)},
MRNUMBER = {3171761},
DOI = {10.4007/annals.2014.179.3.7},
zblnumber = {06302171},
}